On the Singular Values of Generalized Toeplitz Matrices

Abstract.

For a bounded function σ defined on $[0, 1]\times \mathbb{T}$, let $\{s^{(n)}_{k}\}^{n+1}_{k=1}$ be the set of singular values of the (n + 1) x (n + 1) matrix whose (j, k)-entries are equal to

$$ \frac{1}{2\pi}\int^{2\pi}_{0} \sigma(\frac{k}{n},e^{i\theta})e^{-i(j-k)\theta}d\theta,\qquad j,k = 0, 1,...,n. $$

These matrices can be thought of as variable-coefficient Toeplitz matrices or generalized Toeplitz matrices. Matrices of the above form can be also thought of as the discrete analogue of pseudodifferential operators. Under a certain smoothness assumption on the function σ, we prove that

$$ \sum_{k=1}^{n+1} f(s^{2}_{k})\quad = \quad c_{1}\cdot (n+1) + c_{2} + o(1)\qquad as \enskip n \rightarrow \infty, $$

where the constant c ₁ and a part of c ₂ are shown to have explicit integral representations. The other part of c ₂ turns out to have a resemblance to the Toeplitz case. This asymptotic formula can be viewed as a generalization of the classical theory on singular values of Toeplitz matrices.